On the Complexity of Computing Winning Strategies for Finite Poset Games

This paper is concerned with the complexity of computing winning strategies for poset games. While it is reasonably clear that such strategies can be computed in PSPACE, we give a simple proof of this fact by a reduction to the game of geography. We also show how to formalize the reasoning about poset games in Skelley’s theory W₁¹\mathbf{W}_{1}^{1} for PSPACE reasoning. We conclude that W₁¹\mathbf{W}_{1}^{1} can use the “strategy stealing argument” to prove that in poset games with a supremum the first player always has a winning strategy.     

A survey of partial-observation stochastic parity games

We consider two-player zero-sum stochastic games on graphs with ω-regular winning conditions specified as parity objectives. These games have applications in the design and control of reactive systems. We survey the complexity results for the problem of deciding the winner in such games, and in classes of interest obtained as special cases, based on the information and the power of randomization available to the players, on the class of objectives and on the winning mode. On the basis of information, these games can be classified as follows: (a) partial-observation (both players have partial view of the game); (b) one-sided partial-observation (one player has partial-observation and the other player has complete-observation); and (c) complete-observation (both players have complete view of the game). The one-sided partial-observation games have two important subclasses: the one-player games, known as partial-observation Markov decision processes (POMDPs), and the blind one-player games, known as probabilistic automata. On the basis of randomization, (a) the players may not be allowed to use randomization (pure strategies), or (b) they may choose a probability distribution over actions but the actual random choice is external and not visible to the player (actions invisible), or (c) they may use full randomization. Finally, various classes of games are obtained by restricting the parity objective to a reachability, safety, Büchi, or coBüchi condition. We also consider several winning modes, such as sure-winning (i.e., all outcomes of a strategy have to satisfy the winning condition), almost-sure winning (i.e., winning with probability 1), limit-sure winning (i.e., winning with probability arbitrarily close to 1), and value-threshold winning (i.e., winning with probability at least ν, where ν is a given rational).     

Concurrent reachability games

We consider concurrent two-player games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zero-sum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type-1 states, player 1 has a deterministic strategy to always reach the target. From type-2 states, player 1 has a randomized strategy to reach the target with probability 1. From type-3 states, player 1 has for every real ε>0$\epsilon >0$ a randomized strategy to reach the target with probability greater than 1−ε$1-\epsilon$. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type-1 states in linear time, and type-2 and type-3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.     

The complexity of stochastic games

We consider the complexity of stochastic games—simple games of chance played by two players. We show that the problem of deciding which player has the greatest chance of winning the game is in the class NP co-NP.     

The complexity of constraint satisfaction games and QCSP

We study the complexity of two-person constraint satisfaction games. An instance of such a game is given by a collection of constraints on overlapping sets of variables, and the two players alternately make moves assigning values from a finite domain to the variables, in a specified order. The first player tries to satisfy all constraints, while the other tries to break at least one constraint; the goal is to decide whether the first player has a winning strategy. We show that such games can be conveniently represented by a logical form of quantified constraint satisfaction, where an instance is given by a first-order sentence in which quantifiers alternate and the quantifier-free part is a conjunction of (positive) atomic formulas; the goal is to decide whether the sentence is true.While the problem of deciding such a game is PSPACE-complete in general, by restricting the set of allowed constraint predicates, one can obtain infinite classes of constraint satisfaction games of lower complexity. We use the quantified constraint satisfaction framework to study how the complexity of deciding such a game depends on the parameter set of allowed predicates. With every predicate, one can associate certain predicate-preserving operations, called polymorphisms. We show that the complexity of our games is determined by the surjective polymorphisms of the constraint predicates. We illustrate how this result can be used by identifying the complexity of a wide variety of constraint satisfaction games.     

Note on winning positions on pushdown games with ω-regular conditions

We consider infinite two-player games on pushdown graphs. For parity winning conditions, we show that the set of winning positions of each player is regular and we give an effective construction of an alternating automaton recognizing it. This provides a DEXPTIME procedure to decide whether a position is winning for a given player. Finally, using the same methods, we show, for any ω-regular winning condition, that the set of winning positions for a given player is regular and effective.     

Decision algorithms for multiplayer noncooperative games of incomplete information

Extending the complexity results of Reif [1,2] for two player games of incomplete information, this paper (see also [3]) presents algorithms for deciding the outcome for various classes of multiplayer games of incomplete information, i.e., deciding whether or not a team has a winning strategy for a particular game. Our companion paper, [4] shows that these algorithms are indeed asymptotically optimal by providing matching lower bounds. The classes of games to which our algorithms are applicable include games which were not previously known to be decidable. We apply our algorithms to provide alternative upper bounds, and new time-space trade-offs on the complexity of multiperson alternating Turing machines [3]. We analyze the algorithms to characterize the space complexity of multiplayer games in terms of the complexity of deterministic computation on Turing machines.In hierarchical multiplayer games, each additional clique (subset of players with the same information) increases the complexity of the outcome problem by a further exponential. We show that an S(n) space bounded k-player game of incomplete information has a deterministic time upper bound of k + 1 repeated exponentials of S(n). Furthermore, S(n) space bounded k-player blindfold games have a deterministic space upper bound of k repeated exponentials of S(n). This paper proves that this exponential blow-up can occur.We also show that time bounded games do not exhibit such hierarchy. A T(n) time bounded blindfold multiplayer game, as well as a T(n) time bounded multiplayer game of incomplete information, has a deterministic space bound of T(n).     

Pushdown Processes: Games and Model-Checking

A pushdown game is a two player perfect information infinite game on a transition graph of a pushdown automaton. A winning condition in such a game is defined in terms of states appearing infinitely often in the play. It is shown that if there is a winning strategy in a pushdown game then there is a winning strategy realized by a pushdown automaton. An EXPTIME procedure for finding a winner in a pushdown game is presented. The procedure is then used to solve the model-checking problem for the pushdown processes and the propositional μ-calculus. The problem is shown to be DEXPTIME-complete.     

Games with Uniqueness Properties

For two-player games of perfect information such as Checkers, Chess, and Go we introduce uniqueness properties. A game position has a uniqueness property if a winning strategy—should one exist—is forced to be unique. Depending on the way that winning strategy is forced, a uniqueness property is classified as weak, strong, or global. We prove that any reasonable two-player game G is extendable to a game G ^* with the strong uniqueness property for both players, so that, e.g., QBF remains PSPACE-complete under this reduction. For global uniqueness, we introduce a simple game over Boolean formulas with this property, and prove that any reasonable two-player game with the global uniqueness property is reducible to it. We show that the class of languages that reduce to globally unique games equals Niedermeier and Rossmaniths unambiguous alternation class UAP, which is in an interesting region between FewP and SPP.     

The bounded core for games with restricted cooperation

A game with restricted (incomplete) cooperation is a triple (N, v, Ω), where N represents a finite set of players, Ω ? 2^N is a set of feasible coalitions such that N ∈ Ω, and v: Ω → R denotes a characteristic function. Unlike the classical TU games, the core of a game with restricted cooperation can be unbounded. Recently Grabisch and Sudhölter [9] proposed a new solution concept—the bounded core—that associates a game (N, v,Ω) with the union of all bounded faces of the core. The bounded core can be empty even if the core is nonempty. This paper gives two axiomatizations of the bounded core. The first axiomatization characterizes the bounded core for the class G^r of all games with restricted cooperation, whereas the second one for the subclass G_bc^r ? G^r of the games with nonempty bounded cores.     

Computational Aspects of Uncertainty Profiles and Angel-Daemon Games

We analyze the complexity of equilibria problems for a class of strategic zero-sum games, called angel-daemon games. Those games were introduced to asses the performance of the execution of a web orchestration on a moderate faulty or under stress environment. Angel-daemon games are a natural example of zero-sum games whose representation is naturally succinct. We show that the problems of deciding the existence of a pure Nash equilibrium or of a dominant strategy for a given player are ${\Sigma}^{p}_{2}$ -complete. Furthermore, computing the value of an angel-daemon game is EXP-complete. Thus, our results match the already known classification of the corresponding problems for the generic families of succinctly represented games with exponential number of actions.     

Graphical Congestion Games

We consider congestion games with linear latency functions in which each player is aware only of a subset of all the other players. This is modeled by means of a social knowledge graph G in which nodes represent players and there is an edge from i to j if i knows j. Under the assumption that the payoff of each player is affected only by the strategies of the adjacent ones, we first give a complete characterization of the games possessing pure Nash equilibria. Namely, if the social graph G is undirected, the game is an exact potential game and thus isomorphic to a classical congestion game. As a consequence, it always converges and possesses Nash equilibria. On the other hand, if G is directed an equilibrium is not guaranteed to exist, but the game is always convergent and an equilibrium can be found in polynomial time if G is acyclic, even if finding the best equilibrium remains an intractable problem.     

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist.     

Qualitative reachability in stochastic BPA games

We consider a class of infinite-state stochastic games generated by stateless pushdown automata (or, equivalently, 1-exit recursive state machines), where the winning objective is specified by a regular set of target configurations and a qualitative probability constraint ‘>0’ or ‘=1’. The goal of one player is to maximize the probability of reaching the target set so that the constraint is satisfied, while the other player aims at the opposite. We show that the winner in such games can be determined in P for the ‘>0’ constraint, and in NP∩co-NP for the ‘=1’ constraint. Further, we prove that the winning regions for both players are regular, and we design algorithms which compute the associated finite-state automata. Finally, we show that winning strategies can be synthesized effectively.     

Complete problems for deterministic polynomial time

Log space reducibility allows a meaningful study of complexity and completeness for the class $\text{P}$ of problems solvable in polynomial time (as a function of problem size). If any one complete problem for $\text{P}$ is recognizable in log^k(n) space (for a fixed k), or requires at least n^c space (where c depends upon the program), then all complete problems in $\text{P}$ have the same property. A variety of familiar problems are shown complete for $\text{P}$, including context-free emptiness, infiniteness and membership, establishing inconsistency of propositional formulas by unit resolution, deciding whether a player in a two-person game has a winning strategy, and determining whether an element is generated from a set by a binary operation.     

The Complexity of Solving Reachability Games Using Value and Strategy Iteration

Two standard algorithms for approximately solving two-player zero-sum concurrent reachability games are value iteration and strategy iteration. We prove upper and lower bounds of \(2^{m^{\varTheta(N)}}\) on the worst case number of iterations needed by both of these algorithms for providing non-trivial approximations to the value of a game with N non-terminal positions and m actions for each player in each position. In particular, both algorithms have doubly-exponential complexity. Even when the game given as input has only one non-terminal position, we prove an exponential lower bound on the worst case number of iterations needed to provide non-trivial approximations.     

Bandwidth and pebbling

The main results of this paper establish relationships between the bandwidth of a graphG — which is the minimum over all layouts ofG in a line of the maximum distance between images of adjacent vertices ofG — and the ease of playing various pebble games onG. Three pebble games on graphs are considered: the well-known computational pebble game, the “progressive” (i.e., no recomputation allowed) version of the computational pebble game, both of which are played on directed acyclic graphs, and the quite different “breadth-first” pebble game, that is played on undirected graphs. We consider two costs of a play of a pebble game: the minimum number of pebbles needed to play the game on the graphG, and the maximumlifetime of any pebble in the game, i.e., the maximum number of moves that any pebble spends on the graph. The first set of results of the paper prove that the minimum lifetime cost of a play of either of the second two pebble games on a graphG is precisely the bandwidth ofG. The second set of results establish bounds on the pebble demand of all three pebble games in terms of the bandwidth of the graph being pebbled; for instance, the number of pebbles needed to pebble a graphG of bandwidthk is at most min (2k ²+k+1, 2k log₂|G|); and, in addition, there are bandwidth-k graphs that require 3k?1 pebbles. The third set of results relate the difficulty of deciding the cost of playing a pebble game on a given input graphG to the bandwidth ofG; for instance, the Pebble Demand problem forn-vertex graphs of bandwidthf(n) is in the class NSPACE (f(n) log² n); and the Optimal Lifetime Problem for either of the second two pebble games is NP-complete.     

On the effect of quantum noise in a quantum prisoner’s dilemma cellular automaton

The disrupting effect of quantum noise on the dynamics of a spatial quantum formulation of the iterated prisoner’s dilemma game with variable entangling is studied in this work. The game is played in the cellular automata manner, i.e., with local and synchronous interaction. It is concluded in this article that quantum noise induces in fair games the need for higher entanglement in order to make possible the emergence of the strategy pair (Q, Q), which produces the same payoff of mutual cooperation. In unfair quantum versus classic player games, quantum noise delays the prevalence of the quantum player.     

Safety first: a two-stage algorithm for the synthesis of reactive systems

In the game-theoretic approach to the synthesis of reactive systems, specifications are often expressed as ω-regular languages. Computing a winning strategy to an infinite game whose winning condition is an ω-regular language is then the main step in obtaining an implementation. Conjoining all the properties of a specification to obtain a monolithic game suffers from the doubly exponential determinization that is required. Despite the success of symbolic algorithms, the monolithic approach is not practical. Existing techniques achieve efficiency by imposing restrictions on the ω-regular languages they deal with. In contrast, we present an approach that achieves improvement in performance through the decomposition of the problem while still accepting the full set of ω-regular languages. Each property is translated into a deterministic ω-regular automaton explicitly while the two-player game defined by the collection of automata is played symbolically. Safety and persistence properties usually make up the majority of a specification. We take advantage of this by solving the game incrementally. Each safety and persistence property is used to gradually construct the parity game. Optimizations are applied after each refinement of the graph. This process produces a compact symbolic encoding of the parity game. We then compose the remaining properties and solve one final game after possibly solving smaller games to further optimize the graph. An implementation is finally derived from the winning strategies computed. We compare the results of our tool to those of the synthesis tool Anzu.